Ratios & Proportional Relationships Progressions Highlights

Overview Changes from current practice Connections to previous and subsequent grades Surprises Conclusion

Why should you listen to our presentation?

Changes: Not solving proportional equations in 6th grade In sixth grade, the focus is on equivalent ratios, not proportions and solving proportions. The latter is best saved as the culmination of a lot of productive struggle in the sixth grade.

Changes: Not Cross-multiplying until seventh grade Timmy: “Mommy, remember how last year the bullies in seventh grade kicked sand in my face and told me I couldn’t cross-multiply? Well, this year I can cross-multiply AND represent equivalent ratios as equations in two variables with a constant of proportionality! Also, today is stand up for gingers day so everybody was nice to me!”

Visual Models: Tape Diagram This diagram represents a mixture of apple and grape juice with a ratio of 3 to 2.

Visual Models: Equivalent Ratios Vs. Equivalent Fractions

Visual Models: Double Number Line

How the ideas grow from grade to grade: Where they are coming from Use the multiplication table to focus on situations that involve rows and columns. Focus on the rows and columns of a ratio table as multiples of each other Skip counting to extend tables and fill in missing entries Fraction bar representation introduced in elementary school

How the ideas grow from grade to grade: Where they are going Direct variation extends into 8th grade Apply proportions in context for scale drawings Foundation for study of functions, specifically linear functions Extend understanding of quantity to new units like miles per hour instead of 3 miles in one hour

Surprises, Part I

Surprises Part II Introduce constant of proportionality in 7th grade (as opposed to 8th) The idea of a ratio is distinguished from a fraction, all terms have precise definitions which may be different from definitions used by certain textbooks Tape diagrams and double number line - new terminology and alternate representations of ratios

Conclusion The focus for teaching ratios and proportionss in 6th grade is visual and intuitive, rather than computational. The focus in 7th grade is to represent proportional relationships with equations with visual represen- tations of graphs and tables. These strategies seem practical for students and help them to develop number sense and reasoning skills more than the current strategies used.